The linear operators studied by King et al. [1]} and Huber et al. [2]} have been extended to 2D through an identical methodology; time derivatives are kept the same, and spatial derivatives are replaced with 2D del operators. For both operators, we find that an increase in the dimensionality leads to a decrease in growth rates and an increase in decay rates. The changes are equivalent for both operators, and scale the response character by a factor of \(t^{{-1}{2}}\) .
The algebraic growth of \(t^{{1}{2}}\)  at the peak velocities in both [1]}
and in [2]} is reduced to a constant height proportional to \(t^0\) . Along all the other velocities, the behaviour is reduced from \((t^{{-1}{2}})\)  and \((e^{{-\hat{V}}{8}}t^{{-1}{2}})\)  to \((t^{-1})\)  and \((e^{{-\hat{V}}{8}}t^{-1})\)  respectively.
The propagation of wave features in 2-D expand as \(t^{{1}{2}}\) , which is slightly more slowly than the corresponding 1D case for the KRK operator.
Both solutions can be expressed in terms of the similarity variable \(\Phi \)  (in 2D-KRK, for all time; in 2D-CMH, only for large time) which couples time and the radial velocity relative to the peak and allows for relevant features of the solutions to be extracted.
